Integrand size = 32, antiderivative size = 182 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {2 B (b c-a d)^4 g^4 x}{5 d^4}-\frac {B (b c-a d)^3 g^4 (a+b x)^2}{5 b d^3}+\frac {2 B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b}-\frac {2 B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5} \]
2/5*B*(-a*d+b*c)^4*g^4*x/d^4-1/5*B*(-a*d+b*c)^3*g^4*(b*x+a)^2/b/d^3+2/15*B *(-a*d+b*c)^2*g^4*(b*x+a)^3/b/d^2-1/10*B*(-a*d+b*c)*g^4*(b*x+a)^4/b/d+1/5* g^4*(b*x+a)^5*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/b-2/5*B*(-a*d+b*c)^5*g^4*ln( d*x+c)/b/d^5
Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.79 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+\frac {B (b c-a d) \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )}{6 d^5}\right )}{5 b} \]
(g^4*((a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + (B*(b*c - a*d )*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]))/(6 *d^5)))/(5*b)
Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2948, 27, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a g+b g x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2948 |
| \(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B (b c-a d) \int \frac {g^5 (a+b x)^4}{c+d x}dx}{5 b g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 (b c-a d) \int \frac {(a+b x)^4}{c+d x}dx}{5 b}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 (b c-a d) \int \left (\frac {(a d-b c)^4}{d^4 (c+d x)}-\frac {b (b c-a d)^3}{d^4}+\frac {b (a+b x)^3}{d}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (b c-a d)^2 (a+b x)}{d^3}\right )dx}{5 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 (b c-a d) \left (\frac {(b c-a d)^4 \log (c+d x)}{d^5}-\frac {b x (b c-a d)^3}{d^4}+\frac {(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac {(a+b x)^3 (b c-a d)}{3 d^2}+\frac {(a+b x)^4}{4 d}\right )}{5 b}\) |
(g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*b) - (2*B*(b *c - a*d)*g^4*(-((b*(b*c - a*d)^3*x)/d^4) + ((b*c - a*d)^2*(a + b*x)^2)/(2 *d^3) - ((b*c - a*d)*(a + b*x)^3)/(3*d^2) + (a + b*x)^4/(4*d) + ((b*c - a* d)^4*Log[c + d*x])/d^5))/(5*b)
3.2.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*( (A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d)/(g*(m + 1))) Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] / ; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && !(EqQ[m, -2] && IntegerQ[n])
Leaf count of result is larger than twice the leaf count of optimal. \(445\) vs. \(2(170)=340\).
Time = 1.20 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.45
| method | result | size |
| risch | \(-\frac {4 g^{4} b B \,a^{3} c x}{d}+\frac {4 g^{4} b^{2} B \,a^{2} c^{2} x}{d^{2}}-\frac {2 g^{4} b^{3} B a \,c^{3} x}{d^{3}}-\frac {4 g^{4} b^{2} B \ln \left (d x +c \right ) a^{2} c^{3}}{d^{3}}+\frac {2 g^{4} b^{3} B \ln \left (d x +c \right ) a \,c^{4}}{d^{4}}+\frac {4 g^{4} b B \ln \left (d x +c \right ) a^{3} c^{2}}{d^{2}}+\frac {\left (b x +a \right )^{5} g^{4} B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{5 b}-\frac {2 g^{4} b^{3} B a c \,x^{3}}{3 d}-\frac {2 g^{4} b^{2} B \,a^{2} c \,x^{2}}{d}+\frac {g^{4} b^{3} B a \,c^{2} x^{2}}{d^{2}}+\frac {2 g^{4} b^{4} B \,c^{2} x^{3}}{15 d^{2}}+\frac {8 g^{4} b^{2} B \,a^{2} x^{3}}{15}+2 g^{4} b A \,a^{3} x^{2}+\frac {6 g^{4} b B \,a^{3} x^{2}}{5}-\frac {g^{4} b^{4} B \,c^{3} x^{2}}{5 d^{3}}+g^{4} A \,a^{4} x +\frac {8 g^{4} B \,a^{4} x}{5}+\frac {2 g^{4} b^{4} B \,c^{4} x}{5 d^{4}}-\frac {2 g^{4} B \ln \left (d x +c \right ) a^{4} c}{d}-\frac {2 g^{4} b^{4} B \ln \left (d x +c \right ) c^{5}}{5 d^{5}}+\frac {g^{4} b^{4} A \,x^{5}}{5}+g^{4} b^{3} A a \,x^{4}+\frac {g^{4} b^{3} B a \,x^{4}}{10}-\frac {g^{4} b^{4} B c \,x^{4}}{10 d}+2 g^{4} b^{2} A \,a^{2} x^{3}+\frac {2 g^{4} B \ln \left (d x +c \right ) a^{5}}{5 b}\) | \(446\) |
| parallelrisch | \(\frac {6 B \,x^{5} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{5} c \,d^{5} g^{4}+6 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{5} c^{6} g^{4}+12 B \ln \left (b x +a \right ) a^{6} c \,d^{5} g^{4}-12 B \ln \left (b x +a \right ) a \,b^{5} c^{6} g^{4}+4 B \,x^{3} a \,b^{5} c^{3} d^{3} g^{4}+60 A \,x^{2} a^{4} b^{2} c \,d^{5} g^{4}+36 B \,x^{2} a^{4} b^{2} c \,d^{5} g^{4}-60 B \,x^{2} a^{3} b^{3} c^{2} d^{4} g^{4}+30 B \,x^{2} a^{2} b^{4} c^{3} d^{3} g^{4}-6 B \,x^{2} a \,b^{5} c^{4} d^{2} g^{4}+30 A x \,a^{5} b c \,d^{5} g^{4}+48 B x \,a^{5} b c \,d^{5} g^{4}-120 B x \,a^{4} b^{2} c^{2} d^{4} g^{4}+120 B x \,a^{3} b^{3} c^{3} d^{3} g^{4}+6 A \,x^{5} a \,b^{5} c \,d^{5} g^{4}+30 A \,x^{4} a^{2} b^{4} c \,d^{5} g^{4}+3 B \,x^{4} a^{2} b^{4} c \,d^{5} g^{4}-3 B \,x^{4} a \,b^{5} c^{2} d^{4} g^{4}+60 A \,x^{3} a^{3} b^{3} c \,d^{5} g^{4}+16 B \,x^{3} a^{3} b^{3} c \,d^{5} g^{4}-20 B \,x^{3} a^{2} b^{4} c^{2} d^{4} g^{4}+30 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{5} b \,c^{2} d^{4} g^{4}-60 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{4} b^{2} c^{3} d^{3} g^{4}+60 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{3} b^{3} c^{4} d^{2} g^{4}-30 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{2} b^{4} c^{5} d \,g^{4}-60 B \ln \left (b x +a \right ) a^{5} b \,c^{2} d^{4} g^{4}+120 B \ln \left (b x +a \right ) a^{4} b^{2} c^{3} d^{3} g^{4}-120 B \ln \left (b x +a \right ) a^{3} b^{3} c^{4} d^{2} g^{4}+30 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{2} b^{4} c \,d^{5} g^{4}+60 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{3} b^{3} c \,d^{5} g^{4}+60 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{4} b^{2} c \,d^{5} g^{4}+30 B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{5} b c \,d^{5} g^{4}+60 B \ln \left (b x +a \right ) a^{2} b^{4} c^{5} d \,g^{4}-60 B x \,a^{2} b^{4} c^{4} d^{2} g^{4}+12 B x a \,b^{5} c^{5} d \,g^{4}}{30 a b c \,d^{5}}\) | \(839\) |
| parts | \(\text {Expression too large to display}\) | \(1008\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1172\) |
| default | \(\text {Expression too large to display}\) | \(1172\) |
-4*g^4/d*b*B*a^3*c*x+4*g^4/d^2*b^2*B*a^2*c^2*x-2*g^4/d^3*b^3*B*a*c^3*x-4*g ^4/d^3*b^2*B*ln(d*x+c)*a^2*c^3+2*g^4/d^4*b^3*B*ln(d*x+c)*a*c^4+4*g^4/d^2*b *B*ln(d*x+c)*a^3*c^2+1/5*(b*x+a)^5*g^4*B/b*ln(e*(b*x+a)^2/(d*x+c)^2)-2/3*g ^4/d*b^3*B*a*c*x^3-2*g^4/d*b^2*B*a^2*c*x^2+g^4/d^2*b^3*B*a*c^2*x^2+2/15*g^ 4/d^2*b^4*B*c^2*x^3+8/15*g^4*b^2*B*a^2*x^3+2*g^4*b*A*a^3*x^2+6/5*g^4*b*B*a ^3*x^2-1/5*g^4/d^3*b^4*B*c^3*x^2+g^4*A*a^4*x+8/5*g^4*B*a^4*x+2/5*g^4/d^4*b ^4*B*c^4*x-2*g^4/d*B*ln(d*x+c)*a^4*c-2/5*g^4/d^5*b^4*B*ln(d*x+c)*c^5+1/5*g ^4*b^4*A*x^5+g^4*b^3*A*a*x^4+1/10*g^4*b^3*B*a*x^4-1/10*g^4/d*b^4*B*c*x^4+2 *g^4*b^2*A*a^2*x^3+2/5*g^4/b*B*ln(d*x+c)*a^5
Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (170) = 340\).
Time = 0.33 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.49 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {6 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) - 3 \, {\left (B b^{5} c d^{4} - {\left (10 \, A + B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} + 4 \, {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + {\left (15 \, A + 4 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} - 6 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 2 \, {\left (5 \, A + 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} + 6 \, {\left (2 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 20 \, B a^{2} b^{3} c^{2} d^{3} - 20 \, B a^{3} b^{2} c d^{4} + {\left (5 \, A + 8 \, B\right )} a^{4} b d^{5}\right )} g^{4} x - 12 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} \log \left (d x + c\right ) + 6 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{30 \, b d^{5}} \]
1/30*(6*A*b^5*d^5*g^4*x^5 + 12*B*a^5*d^5*g^4*log(b*x + a) - 3*(B*b^5*c*d^4 - (10*A + B)*a*b^4*d^5)*g^4*x^4 + 4*(B*b^5*c^2*d^3 - 5*B*a*b^4*c*d^4 + (1 5*A + 4*B)*a^2*b^3*d^5)*g^4*x^3 - 6*(B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^3 + 1 0*B*a^2*b^3*c*d^4 - 2*(5*A + 3*B)*a^3*b^2*d^5)*g^4*x^2 + 6*(2*B*b^5*c^4*d - 10*B*a*b^4*c^3*d^2 + 20*B*a^2*b^3*c^2*d^3 - 20*B*a^3*b^2*c*d^4 + (5*A + 8*B)*a^4*b*d^5)*g^4*x - 12*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3 *d^2 - 10*B*a^3*b^2*c^2*d^3 + 5*B*a^4*b*c*d^4)*g^4*log(d*x + c) + 6*(B*b^5 *d^5*g^4*x^5 + 5*B*a*b^4*d^5*g^4*x^4 + 10*B*a^2*b^3*d^5*g^4*x^3 + 10*B*a^3 *b^2*d^5*g^4*x^2 + 5*B*a^4*b*d^5*g^4*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e )/(d^2*x^2 + 2*c*d*x + c^2)))/(b*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (163) = 326\).
Time = 3.61 (sec) , antiderivative size = 998, normalized size of antiderivative = 5.48 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {A b^{4} g^{4} x^{5}}{5} + \frac {2 B a^{5} g^{4} \log {\left (x + \frac {\frac {2 B a^{6} d^{5} g^{4}}{b} + 10 B a^{5} c d^{4} g^{4} - 20 B a^{4} b c^{2} d^{3} g^{4} + 20 B a^{3} b^{2} c^{3} d^{2} g^{4} - 10 B a^{2} b^{3} c^{4} d g^{4} + 2 B a b^{4} c^{5} g^{4}}{2 B a^{5} d^{5} g^{4} + 10 B a^{4} b c d^{4} g^{4} - 20 B a^{3} b^{2} c^{2} d^{3} g^{4} + 20 B a^{2} b^{3} c^{3} d^{2} g^{4} - 10 B a b^{4} c^{4} d g^{4} + 2 B b^{5} c^{5} g^{4}} \right )}}{5 b} - \frac {2 B c g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right ) \log {\left (x + \frac {12 B a^{5} c d^{4} g^{4} - 20 B a^{4} b c^{2} d^{3} g^{4} + 20 B a^{3} b^{2} c^{3} d^{2} g^{4} - 10 B a^{2} b^{3} c^{4} d g^{4} + 2 B a b^{4} c^{5} g^{4} - 2 B a c g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right ) + \frac {2 B b c^{2} g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right )}{d}}{2 B a^{5} d^{5} g^{4} + 10 B a^{4} b c d^{4} g^{4} - 20 B a^{3} b^{2} c^{2} d^{3} g^{4} + 20 B a^{2} b^{3} c^{3} d^{2} g^{4} - 10 B a b^{4} c^{4} d g^{4} + 2 B b^{5} c^{5} g^{4}} \right )}}{5 d^{5}} + x^{4} \left (A a b^{3} g^{4} + \frac {B a b^{3} g^{4}}{10} - \frac {B b^{4} c g^{4}}{10 d}\right ) + x^{3} \cdot \left (2 A a^{2} b^{2} g^{4} + \frac {8 B a^{2} b^{2} g^{4}}{15} - \frac {2 B a b^{3} c g^{4}}{3 d} + \frac {2 B b^{4} c^{2} g^{4}}{15 d^{2}}\right ) + x^{2} \cdot \left (2 A a^{3} b g^{4} + \frac {6 B a^{3} b g^{4}}{5} - \frac {2 B a^{2} b^{2} c g^{4}}{d} + \frac {B a b^{3} c^{2} g^{4}}{d^{2}} - \frac {B b^{4} c^{3} g^{4}}{5 d^{3}}\right ) + x \left (A a^{4} g^{4} + \frac {8 B a^{4} g^{4}}{5} - \frac {4 B a^{3} b c g^{4}}{d} + \frac {4 B a^{2} b^{2} c^{2} g^{4}}{d^{2}} - \frac {2 B a b^{3} c^{3} g^{4}}{d^{3}} + \frac {2 B b^{4} c^{4} g^{4}}{5 d^{4}}\right ) + \left (B a^{4} g^{4} x + 2 B a^{3} b g^{4} x^{2} + 2 B a^{2} b^{2} g^{4} x^{3} + B a b^{3} g^{4} x^{4} + \frac {B b^{4} g^{4} x^{5}}{5}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )} \]
A*b**4*g**4*x**5/5 + 2*B*a**5*g**4*log(x + (2*B*a**6*d**5*g**4/b + 10*B*a* *5*c*d**4*g**4 - 20*B*a**4*b*c**2*d**3*g**4 + 20*B*a**3*b**2*c**3*d**2*g** 4 - 10*B*a**2*b**3*c**4*d*g**4 + 2*B*a*b**4*c**5*g**4)/(2*B*a**5*d**5*g**4 + 10*B*a**4*b*c*d**4*g**4 - 20*B*a**3*b**2*c**2*d**3*g**4 + 20*B*a**2*b** 3*c**3*d**2*g**4 - 10*B*a*b**4*c**4*d*g**4 + 2*B*b**5*c**5*g**4))/(5*b) - 2*B*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d**2 - 5*a* b**3*c**3*d + b**4*c**4)*log(x + (12*B*a**5*c*d**4*g**4 - 20*B*a**4*b*c**2 *d**3*g**4 + 20*B*a**3*b**2*c**3*d**2*g**4 - 10*B*a**2*b**3*c**4*d*g**4 + 2*B*a*b**4*c**5*g**4 - 2*B*a*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a **2*b**2*c**2*d**2 - 5*a*b**3*c**3*d + b**4*c**4) + 2*B*b*c**2*g**4*(5*a** 4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d**2 - 5*a*b**3*c**3*d + b** 4*c**4)/d)/(2*B*a**5*d**5*g**4 + 10*B*a**4*b*c*d**4*g**4 - 20*B*a**3*b**2* c**2*d**3*g**4 + 20*B*a**2*b**3*c**3*d**2*g**4 - 10*B*a*b**4*c**4*d*g**4 + 2*B*b**5*c**5*g**4))/(5*d**5) + x**4*(A*a*b**3*g**4 + B*a*b**3*g**4/10 - B*b**4*c*g**4/(10*d)) + x**3*(2*A*a**2*b**2*g**4 + 8*B*a**2*b**2*g**4/15 - 2*B*a*b**3*c*g**4/(3*d) + 2*B*b**4*c**2*g**4/(15*d**2)) + x**2*(2*A*a**3* b*g**4 + 6*B*a**3*b*g**4/5 - 2*B*a**2*b**2*c*g**4/d + B*a*b**3*c**2*g**4/d **2 - B*b**4*c**3*g**4/(5*d**3)) + x*(A*a**4*g**4 + 8*B*a**4*g**4/5 - 4*B* a**3*b*c*g**4/d + 4*B*a**2*b**2*c**2*g**4/d**2 - 2*B*a*b**3*c**3*g**4/d**3 + 2*B*b**4*c**4*g**4/(5*d**4)) + (B*a**4*g**4*x + 2*B*a**3*b*g**4*x**2...
Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (170) = 340\).
Time = 0.22 (sec) , antiderivative size = 885, normalized size of antiderivative = 4.86 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {1}{5} \, A b^{4} g^{4} x^{5} + A a b^{3} g^{4} x^{4} + 2 \, A a^{2} b^{2} g^{4} x^{3} + 2 \, A a^{3} b g^{4} x^{2} + {\left (x \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{4} g^{4} + 2 \, {\left (x^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a^{3} b g^{4} + 2 \, {\left (x^{3} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a^{2} b^{2} g^{4} + \frac {1}{3} \, {\left (3 \, x^{4} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B a b^{3} g^{4} + \frac {1}{30} \, {\left (6 \, x^{5} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b^{4} g^{4} + A a^{4} g^{4} x \]
1/5*A*b^4*g^4*x^5 + A*a*b^3*g^4*x^4 + 2*A*a^2*b^2*g^4*x^3 + 2*A*a^3*b*g^4* x^2 + (x*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2* c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*a*log(b*x + a)/b - 2*c *log(d*x + c)/d)*B*a^4*g^4 + 2*(x^2*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2 ) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - 2*a^2*log(b*x + a)/b^2 + 2*c^2*log(d*x + c)/d^2 - 2*(b*c - a*d)*x/(b*d) )*B*a^3*b*g^4 + 2*(x^3*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x /(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*a^3*log( b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2* c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a^2*b^2*g^4 + 1/3*(3*x^4*log(b^2*e*x^2/(d^2 *x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x ^2 + 2*c*d*x + c^2)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - ( 2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*a*b^3*g^4 + 1/30*(6*x^5*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3* (b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c ^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*b^4*g^4 + A *a^4*g^4*x
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (170) = 340\).
Time = 59.29 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.69 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {1}{5} \, A b^{4} g^{4} x^{5} + \frac {2 \, B a^{5} g^{4} \log \left (b x + a\right )}{5 \, b} - \frac {{\left (B b^{4} c g^{4} - 10 \, A a b^{3} d g^{4} - B a b^{3} d g^{4}\right )} x^{4}}{10 \, d} + \frac {2 \, {\left (B b^{4} c^{2} g^{4} - 5 \, B a b^{3} c d g^{4} + 15 \, A a^{2} b^{2} d^{2} g^{4} + 4 \, B a^{2} b^{2} d^{2} g^{4}\right )} x^{3}}{15 \, d^{2}} + \frac {1}{5} \, {\left (B b^{4} g^{4} x^{5} + 5 \, B a b^{3} g^{4} x^{4} + 10 \, B a^{2} b^{2} g^{4} x^{3} + 10 \, B a^{3} b g^{4} x^{2} + 5 \, B a^{4} g^{4} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {{\left (B b^{4} c^{3} g^{4} - 5 \, B a b^{3} c^{2} d g^{4} + 10 \, B a^{2} b^{2} c d^{2} g^{4} - 10 \, A a^{3} b d^{3} g^{4} - 6 \, B a^{3} b d^{3} g^{4}\right )} x^{2}}{5 \, d^{3}} + \frac {{\left (2 \, B b^{4} c^{4} g^{4} - 10 \, B a b^{3} c^{3} d g^{4} + 20 \, B a^{2} b^{2} c^{2} d^{2} g^{4} - 20 \, B a^{3} b c d^{3} g^{4} + 5 \, A a^{4} d^{4} g^{4} + 8 \, B a^{4} d^{4} g^{4}\right )} x}{5 \, d^{4}} - \frac {2 \, {\left (B b^{4} c^{5} g^{4} - 5 \, B a b^{3} c^{4} d g^{4} + 10 \, B a^{2} b^{2} c^{3} d^{2} g^{4} - 10 \, B a^{3} b c^{2} d^{3} g^{4} + 5 \, B a^{4} c d^{4} g^{4}\right )} \log \left (-d x - c\right )}{5 \, d^{5}} \]
1/5*A*b^4*g^4*x^5 + 2/5*B*a^5*g^4*log(b*x + a)/b - 1/10*(B*b^4*c*g^4 - 10* A*a*b^3*d*g^4 - B*a*b^3*d*g^4)*x^4/d + 2/15*(B*b^4*c^2*g^4 - 5*B*a*b^3*c*d *g^4 + 15*A*a^2*b^2*d^2*g^4 + 4*B*a^2*b^2*d^2*g^4)*x^3/d^2 + 1/5*(B*b^4*g^ 4*x^5 + 5*B*a*b^3*g^4*x^4 + 10*B*a^2*b^2*g^4*x^3 + 10*B*a^3*b*g^4*x^2 + 5* B*a^4*g^4*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2) ) - 1/5*(B*b^4*c^3*g^4 - 5*B*a*b^3*c^2*d*g^4 + 10*B*a^2*b^2*c*d^2*g^4 - 10 *A*a^3*b*d^3*g^4 - 6*B*a^3*b*d^3*g^4)*x^2/d^3 + 1/5*(2*B*b^4*c^4*g^4 - 10* B*a*b^3*c^3*d*g^4 + 20*B*a^2*b^2*c^2*d^2*g^4 - 20*B*a^3*b*c*d^3*g^4 + 5*A* a^4*d^4*g^4 + 8*B*a^4*d^4*g^4)*x/d^4 - 2/5*(B*b^4*c^5*g^4 - 5*B*a*b^3*c^4* d*g^4 + 10*B*a^2*b^2*c^3*d^2*g^4 - 10*B*a^3*b*c^2*d^3*g^4 + 5*B*a^4*c*d^4* g^4)*log(-d*x - c)/d^5
Time = 1.71 (sec) , antiderivative size = 1025, normalized size of antiderivative = 5.63 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{10\,b\,d}+\frac {a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{2\,b\,d}\right )-x^3\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{3\,d}+\frac {A\,a\,b^3\,c\,g^4}{3\,d}\right )+x\,\left (\frac {a^3\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c+4\,B\,a\,d-4\,B\,b\,c\right )}{d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{5\,b\,d}+\frac {2\,a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{b\,d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{b\,d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (B\,a^4\,g^4\,x+2\,B\,a^3\,b\,g^4\,x^2+2\,B\,a^2\,b^2\,g^4\,x^3+B\,a\,b^3\,g^4\,x^4+\frac {B\,b^4\,g^4\,x^5}{5}\right )+x^4\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{20\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (10\,B\,a^4\,c\,d^4\,g^4-20\,B\,a^3\,b\,c^2\,d^3\,g^4+20\,B\,a^2\,b^2\,c^3\,d^2\,g^4-10\,B\,a\,b^3\,c^4\,d\,g^4+2\,B\,b^4\,c^5\,g^4\right )}{5\,d^5}+\frac {A\,b^4\,g^4\,x^5}{5}+\frac {2\,B\,a^5\,g^4\,\ln \left (a+b\,x\right )}{5\,b} \]
x^2*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c) )/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a *b^2*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d + (A*a*b^3*c*g^4)/d)) /(10*b*d) + (a^2*b*g^4*(5*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d - (a*c*( (b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a *d + 5*b*c))/(5*d)))/(2*b*d)) - x^3*((((b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B* a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c) )/(15*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/(3*d) + (A*a*b^3*c*g^4)/(3*d)) + x*((a^3*g^4*(5*A*a*d + 10*A*b*c + 4*B*a*d - 4*B*b *c))/d - ((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b* c + 2*B*a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d + (A*a*b^3*c*g^4)/d))/(5*b*d) + (2*a^2*b*g^4*(5*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d - (a*c*((b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/ (5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d)))/(b*d)))/(5*b*d) + (a*c*((((b^3 *g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d + (A*a*b^3*c*g^4)/d))/(b*d)) + log((e*(a + b*x)^2) /(c + d*x)^2)*((B*b^4*g^4*x^5)/5 + B*a^4*g^4*x + 2*B*a^3*b*g^4*x^2 + B*a*b ^3*g^4*x^4 + 2*B*a^2*b^2*g^4*x^3) + x^4*((b^3*g^4*(25*A*a*d + 5*A*b*c +...